Think of the phrases you use daily when out in public. Learn how to greet people, apologize, and "excuse me." You should also learn how to order food, ask for directions, and ask if they know how to speak English. I think learning how to ask if someone speaks English is the most important phrase to learn because it's extremely rude to go up to someone in a foreign country and immediately speak English to them.

Make sure you know the words/phrases for exchanging pleasantries, ordering food in restaurants, directional terms, and, most importantly, how to ask for help. In addition, you want to know how to ask basic questions (i.e. where the bathroom is) and also you want to be able to somewhat communicate with professors who may not speak great English (i.e. how to ask about grades, deadlines, assignments, etc.).

The most important phrases to know are ones that pertain to your necessities: "where is the bathroom?" and "where is the nearest grocery store?" are good examples. I'll echo Madison and emphasize that you need to be able to greet people and say goodbye, but I'd say you cannot get by without being able to express that this language is not your native language, and you'll want to know phrases like "can you repeat that?" or "would you mind speaking more slowly?" so that you can better understand people around you! Above all, make sure you know how to ask for what you need (food, water, bathroom, etc).

I think that the most important phrases are ones you learn out of politeness. These are common ones like "please," "thank you," "sorry," and "hello/goodbye". When travelling or living in an unknown area, the last thing you want to do is offend somebody. The most surefire way to make sure you don't is by using polite phrases like the ones mentioned above. Besides these, I would learn some common ones like how to say "Where is ____" and one explaining that you only understand English.